If $\sin (x+y) = 3x - 2y$, Then $\frac{dy}{dx} =$A. $\frac{3 - \cos (x+y)}{2}$B. $\frac{1 - \cos (x+y)}{\cos (x+y)}$C. $\frac{3}{2 + \cos (x+y)}$D. $\frac{3 - \cos (x+y)}{2 + \cos (x+y)}$

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If sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, then dydx=\frac{dy}{dx} =: A Comprehensive Analysis

In this article, we will delve into the world of trigonometry and calculus to solve a complex problem involving the derivative of a function. The problem states that sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, and we are asked to find the value of dydx\frac{dy}{dx}. This problem requires a deep understanding of trigonometric identities, differentiation rules, and algebraic manipulations.

The given equation is sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y. To find the value of dydx\frac{dy}{dx}, we need to use the chain rule of differentiation, which states that if we have a composite function of the form f(g(x))f(g(x)), then the derivative of this function is given by f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x). In this case, we can consider f(x)=sin⁡xf(x) = \sin x and g(x)=x+yg(x) = x+y.

Using the chain rule, we can write the derivative of sin⁡(x+y)\sin (x+y) as:

ddxsin⁡(x+y)=cos⁡(x+y)⋅ddx(x+y)\frac{d}{dx} \sin (x+y) = \cos (x+y) \cdot \frac{d}{dx} (x+y)

Now, we can use the fact that the derivative of x+yx+y is 11 to simplify the expression:

ddxsin⁡(x+y)=cos⁡(x+y)⋅1\frac{d}{dx} \sin (x+y) = \cos (x+y) \cdot 1

ddxsin⁡(x+y)=cos⁡(x+y)\frac{d}{dx} \sin (x+y) = \cos (x+y)

Now, we can differentiate the given equation sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y with respect to xx:

ddxsin⁡(x+y)=ddx(3x−2y)\frac{d}{dx} \sin (x+y) = \frac{d}{dx} (3x - 2y)

Using the chain rule, we can write the derivative of sin⁡(x+y)\sin (x+y) as:

cos⁡(x+y)=3−2dydx\cos (x+y) = 3 - 2 \frac{dy}{dx}

Now, we can solve for dydx\frac{dy}{dx} by isolating it on one side of the equation:

2dydx=3−cos⁡(x+y)2 \frac{dy}{dx} = 3 - \cos (x+y)

dydx=3−cos⁡(x+y)2\frac{dy}{dx} = \frac{3 - \cos (x+y)}{2}

In this article, we have used the chain rule of differentiation to solve a complex problem involving the derivative of a function. We have shown that if sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, then dydx=3−cos⁡(x+y)2\frac{dy}{dx} = \frac{3 - \cos (x+y)}{2}. This problem requires a deep understanding of trigonometric identities, differentiation rules, and algebraic manipulations.

The correct answer is:

A. 3−cos⁡(x+y)2\frac{3 - \cos (x+y)}{2}
If sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, then dydx=\frac{dy}{dx} =: A Comprehensive Analysis and Q&A

In our previous article, we delved into the world of trigonometry and calculus to solve a complex problem involving the derivative of a function. The problem states that sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, and we are asked to find the value of dydx\frac{dy}{dx}. In this article, we will provide a comprehensive analysis and answer some frequently asked questions related to this problem.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a fundamental concept in calculus that allows us to differentiate composite functions. It states that if we have a composite function of the form f(g(x))f(g(x)), then the derivative of this function is given by f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x).

Q: How do we apply the chain rule to the given equation?

A: To apply the chain rule, we need to identify the outer and inner functions. In this case, the outer function is sin⁡x\sin x and the inner function is x+yx+y. We can then use the chain rule to write the derivative of sin⁡(x+y)\sin (x+y) as cos⁡(x+y)⋅ddx(x+y)\cos (x+y) \cdot \frac{d}{dx} (x+y).

Q: What is the derivative of x+yx+y?

A: The derivative of x+yx+y is 11, since the derivative of xx is 11 and the derivative of yy is also 11.

Q: How do we differentiate the given equation?

A: To differentiate the given equation, we need to use the chain rule. We can write the derivative of sin⁡(x+y)\sin (x+y) as cos⁡(x+y)⋅1\cos (x+y) \cdot 1, and then equate it to the derivative of 3x−2y3x - 2y, which is 3−2dydx3 - 2 \frac{dy}{dx}.

Q: How do we solve for dydx\frac{dy}{dx}?

A: To solve for dydx\frac{dy}{dx}, we need to isolate it on one side of the equation. We can do this by subtracting 33 from both sides of the equation and then dividing both sides by −2-2.

Q: What is the final answer?

A: The final answer is 3−cos⁡(x+y)2\frac{3 - \cos (x+y)}{2}.

  • Not applying the chain rule correctly
  • Not identifying the outer and inner functions
  • Not differentiating the given equation correctly
  • Not solving for dydx\frac{dy}{dx} correctly
  • Make sure to apply the chain rule correctly
  • Identify the outer and inner functions clearly
  • Differentiate the given equation carefully
  • Solve for dydx\frac{dy}{dx} step by step

In this article, we have provided a comprehensive analysis and answer some frequently asked questions related to the problem of finding the derivative of a function. We have shown that if sin⁡(x+y)=3x−2y\sin (x+y) = 3x - 2y, then dydx=3−cos⁡(x+y)2\frac{dy}{dx} = \frac{3 - \cos (x+y)}{2}. We hope that this article has been helpful in understanding the concept of the chain rule and how to apply it to solve complex problems.